Weibull Analysis Comparing Groups of Data

The RELIABILITY Procedure

Weibull Analysis Comparing Groups of Data

This example illustrates probability plotting and distribution fitting for data grouped by the levels of a special group-variable. The data are from an accelerated life test of an insulating fluid and are the times to electrical breakdown of the fluid under different high voltage levels. Each voltage level defines a subset of data for which a separate analysis and Weibull plot are produced. These data are the 26kV, 30kV, 34kV, and 38kV groups of the data provided by Nelson (1990, p. 129). The following statements create a SAS data set containing the lifetimes and voltages.

   data fluid;                     
      input time voltage$ @@;      
      datalines;                       
         5.79 26kV 1579.52 26kV 
      2323.70 26kV    7.74 30kV 
        17.05 30kV   20.46 30kV 
        21.02 30kV   22.66 30kV 
        43.40 30kV   47.30 30kV 
       139.07 30kV  144.12 30kV 
       175.88 30kV  194.90 30kV 
         0.19 34kV    0.78 34kV 
         0.96 34kV    1.31 34kV 
         2.78 34kV    3.16 34kV 
         4.15 34kV    4.67 34kV 
         4.85 34kV    6.50 34kV 
         7.35 34kV    8.01 34kV 
         8.27 34kV   12.06 34kV 
        31.75 34kV   32.52 34kV 
        33.91 34kV   36.71 34kV 
        72.89 34kV    0.09 38kV 
         0.39 38kV    0.47 38kV 
         0.73 38kV    0.74 38kV 
         1.13 38kV    1.40 38kV 
         2.38 38kV              
         ;                            
   run;

The variable TIME provides the time to breakdown in minutes, and the variable VOLTAGE provides the voltage level at which the test was conducted. These data are not censored.

The RELIABILITY procedure plots the data for the different voltage levels on the same Weibull probability plot, fits a separate distribution to the data at each voltage level, and superimposes distribution lines on the plot.

The following statements produce the probability plot shown in Figure 30.3 for the variable TIME at each level of the group-variable VOLTAGE.

   symbol1 c=red v=plus;
   symbol2 c=blue v=x;
   symbol3 c=green v=square;
   symbol4 c=yellow v=circle;
   proc reliability data=fluid;
      distribution weibull;
      pplot time=voltage  / overlay 
                            noconf
                            cframe = ligr;
   run;

The input data set FLUID is specified by the DATA= option in the PROC RELIABILITY statement. The PROBPLOT statement option OVERLAY specifies that plots for the groups are to be overlaid rather than displayed separately. The option NOCONF specifies that no confidence bands are to be plotted, since these can interfere with one another on overlaid plots; confidence bands are displayed by default.

Figure 30.3: Weibull Probability Plot for the Insulating Fluid Data

A summary table that contains information for all groups is displayed. In addition, information identical to that shown in Figure 30.2 is tabulated for each level of voltage. The summary table for all groups and the tables for the 26kV group are shown in Figure 30.4.

The RELIABILITY Procedure

Model Information - All Groups
Input Data Set	WORK.FLUID
Analysis Variable	time
Distribution	Weibull
Estimation Method	Maximum Likelihood
Confidence Coefficient	95%
Observations Used	41

The RELIABILITY Procedure

	Group
Algorithm converged.	26kv

Summary of Fit
		Group
Observations Used	3	26kv
Uncensored Values	3	26kv
Maximum Loglikelihood	-6.845551	26kv

Weibull Parameter Estimates
Parameter	Estimate	Standard Error	Asymptotic Normal		Group
			95% Confidence Limits
			Lower	Upper
EV Location	6.8625	1.1040	4.6986	9.0264	26kv
EV Scale	1.8342	0.9611	0.6568	5.1226	26kv
Weibull Scale	955.7467	1055.1862	109.7941	8319.6794	26kv
Weibull Shape	0.5452	0.2857	0.1952	1.5226	26kv

Other Weibull Distribution Parameters
Parameter	Value	Group
Mean	1649.4882	26kv
Mode	0.0000	26kv
Median	487.9547	26kv

Weibull Percentile Estimates
Percent	Estimate	Standard Error	Asymptotic Normal		Group
			95% Confidence Limits
			Lower	Upper
0.1	0.00300636	0.02113841	3.11203E-9	2904.27046	26kv
0.2	0.01072998	0.06838144	4.03597E-8	2852.65767	26kv
0.5	0.0577713	0.31803193	1.19079E-6	2802.78862	26kv
1	0.20695478	1.00385021	0.00001538	2784.16263	26kv
2	0.74484901	3.12705686	0.00019885	2790.0941	26kv
5	4.1142692	13.7388263	0.00591379	2862.3304	26kv
10	15.406565	41.4763373	0.07873508	3014.69497	26kv
20	61.0231127	125.020566	1.10053199	3383.65475	26kv
30	144.246801	242.203982	5.36856883	3875.73303	26kv
40	278.770459	398.048692	16.9761581	4577.77125	26kv
50	487.954708	610.02855	42.0948552	5656.26835	26kv
60	814.147288	920.537706	88.770543	7466.84412	26kv
70	1343.42243	1433.97868	165.818889	10884.0666	26kv
80	2287.87124	2445.52431	281.5628	18590.3635	26kv
90	4412.96962	5148.34986	448.419608	43428.7452	26kv
95	7150.89745	9248.2654	566.892142	90202.9338	26kv
99	15735.8513	24666.0388	728.831025	339745.437	26kv
99.9	33104.172	62018.1074	841.826189	1301796.28	26kv

Figure 30.4: Partial Listing of the Tabular Output for the Insulating Fluid Data

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